Existence of traveling breather solutions to cubic nonlinear Maxwell equations in waveguide geometries
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arXiv
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| Format: | Preprint |
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2024
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| author | Ohrem, Sebastian Reichel, Wolfgang |
| author_facet | Ohrem, Sebastian Reichel, Wolfgang |
| contents | We consider the full set of Maxwell equations in a slab or cylindrical waveguide with a cubically nonlinear material law for the polarization of the electric field. The nonlinear polarization may be instantaneous or retarded, and we assume it to be confined inside the core of the waveguide. We prove existence of infinitely many spatially localized, real-valued and time-periodic solutions (breathers) propagating inside the waveguide by applying a variational minimization method to the resulting scalar quasilinear elliptic-hyperbolic equation for the profile of the breathers. The temporal period of the breathers has to be carefully chosen depending on the linear properties of the waveguide. As an example, our results apply if a two-layered linear axisymmetric waveguide is enhanced by a third core region with low refractive index where also the nonlinearity is located. In this case we can also connect our existence result with a bifurcation result. We illustrate our results with numerical simulations. Our solutions are polychromatic functions in general, but for some special models of retarded nonlinear material laws, also monochromatic solutions can exist. In this case the numerical simulations raise an interesting open question: are the breather solutions with minimal energy monochromatic or polychromatic? |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_18729 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of traveling breather solutions to cubic nonlinear Maxwell equations in waveguide geometries Ohrem, Sebastian Reichel, Wolfgang Analysis of PDEs Primary: 35Q61, 49J10, Secondary: 35C07, 78A50 We consider the full set of Maxwell equations in a slab or cylindrical waveguide with a cubically nonlinear material law for the polarization of the electric field. The nonlinear polarization may be instantaneous or retarded, and we assume it to be confined inside the core of the waveguide. We prove existence of infinitely many spatially localized, real-valued and time-periodic solutions (breathers) propagating inside the waveguide by applying a variational minimization method to the resulting scalar quasilinear elliptic-hyperbolic equation for the profile of the breathers. The temporal period of the breathers has to be carefully chosen depending on the linear properties of the waveguide. As an example, our results apply if a two-layered linear axisymmetric waveguide is enhanced by a third core region with low refractive index where also the nonlinearity is located. In this case we can also connect our existence result with a bifurcation result. We illustrate our results with numerical simulations. Our solutions are polychromatic functions in general, but for some special models of retarded nonlinear material laws, also monochromatic solutions can exist. In this case the numerical simulations raise an interesting open question: are the breather solutions with minimal energy monochromatic or polychromatic? |
| title | Existence of traveling breather solutions to cubic nonlinear Maxwell equations in waveguide geometries |
| topic | Analysis of PDEs Primary: 35Q61, 49J10, Secondary: 35C07, 78A50 |
| url | https://arxiv.org/abs/2407.18729 |